Discrete Uniform Distribution Quiz and Flashcards: Mean, Variance, MGF

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the probability mass function (PMF) of a uniform distribution U(a,b)?

$P(X=x)=\frac{1}{2(b-a+1)}$

$P(X=x)=\frac{1}{b-a+1}$ (correct)

$P(X=x)=\frac{1}{b+a-1}$

$P(X=x)=\frac{1}{a+b+1}$

What is the variance of a uniform distribution U(a,b)?

$Var(X)=\frac{(b+a-1)^2}{12}$

$Var(X)=\frac{a+b+1}{12}$

$Var(X)=\frac{(b-a+1)^2}{12}$ (correct)

$Var(X)=\frac{(b-a)^2}{12}$

What is the moment generating function (MGF) of a uniform distribution U(a,b)?

$MX(t)=(a+b+1)(e^t-1)e^{at}-e^{(2b)t}$

$MX(t)=(b-a)(e^t-1)e^{at}-e^{bt}$

$MX(t)=(b+a-1)(e^t-1)e^{at}-e^t$

$MX(t)=(b-a+1)(e^t-1)e^{at}-e^{(b+1)t}$ (correct)

For a uniform distribution U(2,8), what is the mean?

$3$ Signup and view all the answers

What is the variance of the uniform distribution U(4,9)?

$Var(X)=\frac{6^2}{12}$ Signup and view all the answers

Calculate the moment generating function for the uniform distribution U(3,7). What is the result?

$(7-3+1)(e^t-1)e^{3t}-e^{8t}$ Signup and view all the answers

In a uniform distribution U(0,5), what is the probability of X taking a value of 3?

$\frac{1}{5}$ Signup and view all the answers

Find the variance for the uniform distribution U(6,10). What is the correct variance calculation?

$Var(X)=\frac{5^2}{12}$ Signup and view all the answers

In a uniform distribution U(3,9), what is the probability of X being 7?

$\frac{1}{6}$ Signup and view all the answers

Calculate the mean for the uniform distribution U(5,11). What is the correct mean value?

$8$ Signup and view all the answers

What is the mean and variance of a Poisson distribution?

Mean: ?=?, Variance: ?2=? Signup and view all the answers

What is the moment generating function (MGF) of a Binomial distribution?

$MX(t)=e^{(0.7+0.21e^t+0.0153e^{2t})}$ Signup and view all the answers

In the Poisson distribution, what does the parameter ? represent?

Average rate of occurrence of events Signup and view all the answers

What does the Probability Mass Function (PMF) represent in a Binomial distribution?

$P(X=k)=(kn?)p^k(1-p)^{n-k}$ Signup and view all the answers

What is the formula for the moment generating function (MGF) of a Poisson distribution?

$MX(t)=e^{\theta(e^t-1)}$ Signup and view all the answers

What does the binomial coefficient (kn?) represent in a Binomial distribution PMF?

$\frac{n!}{k!(n-k)!}$ Signup and view all the answers

In the context of moment generating functions, what does E[etX] stand for?

$MX(t)$ Signup and view all the answers

Study Notes

Discrete Probability Distributions

Uniform Distribution (Discrete)

Notation: U(a, b) where a and b are the minimum and maximum values of the distribution
Probability Mass Function (PMF): 1/(b-a+1) for x=a, a+1, ..., b, and 0 for other values of x
Mean (μ): (a+b)/2
Variance (σ²): (b-a+1)²/12
Moment Generating Function (MGF): (b-a+1)(e^t-1) / (e^(b+1)t - e^at)

Example: Uniform Distribution U(1, 6)

Mean: 3.5
Variance: 35/12
Moment Generating Function: (6(e^t-1)) / (e^7t - e^t)

Binomial Distribution

Notation: B(n, p) where n is the number of trials and p is the probability of success
Probability Mass Function (PMF): (k choose n) p^k (1-p)^(n-k) where k is the number of successes
Mean (μ): np
Variance (σ²): np(1-p)
Moment Generating Function (MGF): Σ(etk) (k choose n) p^k (1-p)^(n-k) from k=0 to n

Example: Binomial Distribution B(5, 0.3)

Mean: 1.5
Variance: 1.05
Moment Generating Function: (0.7 + 0.21e^t + 0.0153e^(2t))

Poisson Distribution

Notation: Poisson(λ)
Probability Mass Function (PMF): e^(-λ) (λ^k) / k! where k is the number of events
Mean (μ): λ
Variance (σ²): λ
Moment Generating Function (MGF): e^(λ(e^t-1))

Example: Poisson Distribution Poisson(2)

Mean: 2
Variance: 2
Moment Generating Function: e^(2(e^t-1))

Continuous Probability Distributions

Continuous Uniform Distribution

A continuous uniform distribution is a probability distribution where every value between a certain range has an equal likelihood of occurring.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Learn about the properties of the discrete uniform distribution including notation, probability mass function, mean, variance, and moment generating functions. Understand how to calculate the mean and variance of a uniform distribution with minimum (a) and maximum (b) values.

Discrete Uniform Distribution: Mean, Variance, MGF

Choose a study mode